Timeline for finding isothermal coordinates uniformly
Current License: CC BY-SA 2.5
8 events
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| Mar 22, 2011 at 3:02 | comment | added | macbeth | @Michael Beeson: Since $W$ and $V$ are two different parametrizations of $X$, we can write $V=W\circ\varphi$, for some self-diffeomorphism $\varphi$ of the disc. Since $W$ and $V$ are both conformal, we know that $\varphi$ must be locally conformal, hence conformal. | |
| Mar 21, 2011 at 23:38 | comment | added | Michael Beeson | Now I have another difficulty with your answer. Let's check the uniqueness. Suppose $W$ and $V$ are two different isothermal parametrizations of $X$ (taking 0 to the same point and having the same x-derivative at 0). Then if I understand you, you want to define $\varphi:= W^{-1} V$ and say that $\varphi$ is a conformal mapping of the disk, fixing 0 and with $\varphi_x(0) = 1$, and therefore $\varphi$ is the identity. Nice, but, the domain of $W^{-1}$ is a geometric surface, so how do I define it if $W$ is not one-one? It seems this will only work to get LOCAL isothermal coordinates. | |
| Mar 17, 2011 at 16:36 | comment | added | Anton Petrunin | The isothermal coordinate is a solution of some elliptic PDE with coefficients taken from the first fundamental form. Such solutions are known to be uniformly smooth in a compact subdomain. From this you should get $C^\infty$ case. For $C^n$ you have to be more careful, the first fundamental form is in class $C^{n-1}$, but if I remember right solution of elliptic equation with such coefficient should be $C^n$... (It was a while since I study this stuff.) | |
| Mar 16, 2011 at 20:26 | comment | added | Michael Beeson | I tried to work out the details. There is a gap, though. Suppose we have a sequence of isothermal parametrizations $Y^{t_n}$ that are bounded in $C^n$ norm away from $X^0$. Now we need to argue by compactness that some subsequence converges. For that we need a uniform bound on the isothermal parametrizations $Y^t$. Where will we get a bound on the $C^n$ norms of the $Y^t$? | |
| Mar 15, 2011 at 19:57 | comment | added | Michael Beeson | This seems like an excellent idea, and a big improvement over trying to check the detailed proofs. As soon as I can work out the details I will accept your answer. | |
| Mar 15, 2011 at 17:42 | history | edited | Anton Petrunin | CC BY-SA 2.5 |
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| Mar 15, 2011 at 4:02 | history | edited | Anton Petrunin | CC BY-SA 2.5 |
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| Mar 15, 2011 at 2:37 | history | answered | Anton Petrunin | CC BY-SA 2.5 |